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point lies on a line passing through incenter and perpendicular to BC

Source: 2016 Oral Moscow Geometry Olympiad grades 10-11 p1

July 29, 2019

geometryincenter

Problem Statement

The line passing through the center

I

of the inscribed circle of a triangle

ABC

, perpendicular to

AI

and intersects sides

AB

and

AC

at points

C'

and

B'

, respectively. In the triangles

BC'I

and

CB'I

, the altitudes

C'C_1

and

B'B_1

were drawn, respectively. Prove that the midpoint of the segment

B_1C_1

lies on a straight line passing through point

I

and perpendicular to

BC